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A Note on the Equivalence of the Graded Response Model and the Sequential Model

Published online by Cambridge University Press:  01 January 2025

Timo M. Bechger  and
Wies Akkermans
Timo M. Bechger*
Affiliation:
National Institute for Educational Measurement (Cito)
Wies Akkermans
Affiliation:
Centre for Biometry Wageningen
*
Requests for reprints should be sent to National Institute for Educational Measurement, P.O. Box 1034, 6801 MG, Arnhem, THE NETHERLANDS. E-Mail: timo.bechger@cito.nl
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Abstract

This paper concerns items that consist of several item steps to be responded to sequentially. The item scoreX is defined as the number of correct responses until the first failure. Samejima's graded response model states that each step h=1,...,m is characterized by a parameter bh, and, for a subject with ability θ, Pr(X ≥ h; θ) = F(θ − bh). Tutz's general sequential model associates with each step a parameterdh, and it states that Pr(X≥h;θ)=Πr=1hG(θ−dr). Tutz's (1991, 1997) conjectures that the models are equivalent if and only ifF(x)=G(x) is an extreme value distribution. This paper presents a proof for this conjecture.

Keywords

item response theorysequential scoringsequential modelgraded response model
Type
Notes And Comments
Information
Psychometrika , Volume 66 , Issue 3 , September 2001 , pp. 461 - 463
Copyright
Copyright © 2001 The Psychometric Society

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References

Castillo, Ron E. (1996). Algunas applicacionces de las ecuaciones functionales [Some applications of functional equations], Santander, Spain: Universidad de Cantabria.Google Scholar
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Tutz, G. (1997). Sequential models for ordered responses. In van der Linden, W.J., & Hambleton, R. (Eds.), Modern handbook of item response theory (pp. 139–152). New-York, NY: Springer.CrossRefGoogle Scholar